Fixed-background Maxwell presymplectic current

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as inherited from the smooth principal-connection convention. Fix an oriented, time-oriented smooth Lorentzian four-manifold $(M,g,N,o)$ and a fixed normalized $U(1)$ connection sector as in the fixed-topology affine-sector result. Its connection variations are global smooth real one-forms $\alpha ,\beta \in {\mathrm{\Omega }}^1(M)$, with components of unit ${\mathrm{m}}^{-1}$ in length-coordinate charts. Length-coordinate charts use metre-valued coordinates, with $x^0=c{t}_{\mathrm{s}\mathrm{e}\mathrm{c}}$ when time is recorded in seconds; $c$ is a coordinate unit-conversion scale in $\mathrm{m}{\mathrm{s}}^{-1}$, with no numerical measured value or physical role asserted. Impose the fixed-background Maxwell operator equations $d{\star }_{g}d\alpha =0$ and $d{\star }_{g}d\beta =0$, where ${\star }_g$ is the oriented Lorentzian Hodge map of signature $(-,+,+,+)$. Define the real three-form

${\omega }_g(\alpha ,\beta ):=\alpha \wedge {\star }_{g}d\beta -\beta \wedge {\star }_{g}d\alpha \mathrm{.}$

Let ${\iota }_{\mathrm{\Sigma }}:\mathrm{\Sigma }\hookrightarrow M$ be an oriented smooth embedding of a possibly noncompact three-dimensional manifold, and require the induced metric $\gamma :={\iota }_{\mathrm{\Sigma }}^{\ast }g$ to be positive definite. Its oriented Riemannian volume form is the nowhere-vanishing three-form ${\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$ whose expression in each positively oriented chart $y=(y^1,y^2,y^3)$ is $\sqrt{\det ({\gamma }_{ij})}d{y}^1\wedge d{y}^2\wedge d{y}^3$. Let ${\mu }_{\gamma }$ be the positive smooth density measure with coordinate expression $d{\mu }_{\gamma }=\sqrt{\det ({\gamma }_{ij})}{d}^{3}y$. The density-measure integration supplier listed below gives the chart-independent measure and integral; for a real $L^1({\mu }_{\gamma })$ function its signed integral is the finite difference of the integrals of its positive and negative parts, and its chart formula is absolutely convergent.

There is a unique smooth real function $u_{\alpha ,\beta }$ on $\mathrm{\Sigma }$ such that ${\iota }_{\mathrm{\Sigma }}^{\ast }{\omega }_g(\alpha ,\beta )=u_{\alpha ,\beta }{\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$. Define the pairing ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta )$ precisely when $u_{\alpha ,\beta }\in L^1(\mathrm{\Sigma },{\mu }_{\gamma })$, meaning ${\int }_{\mathrm{\Sigma }}\mathrm{\mid }{u}_{\alpha ,\beta }\mathrm{\mid }d{\mu }_{\gamma }<\mathrm{\infty }$, and then set ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta ):={\int }_{\mathrm{\Sigma }}{u}_{\alpha ,\beta }d{\mu }_{\gamma }$. This absolute-integrability condition is the convergence convention on noncompact $\mathrm{\Sigma }$ and makes the value independent of any exhaustion or ordering. In length coordinates on $\mathrm{\Sigma }$, the coefficients of $d\alpha$ and $d\beta$ have units ${\mathrm{m}}^{-2}$, those of ${\omega }_g$ have units ${\mathrm{m}}^{-3}$, $u_{\alpha ,\beta }$ has units ${\mathrm{m}}^{-3}$, and $d{\mu }_{\gamma }$ has volume units ${\mathrm{m}}^3$; thus the normalized integral is dimensionless. Replacing normalized variations by physical-potential variations ${\kappa }_0\alpha$ and ${\kappa }_0\beta$ multiplies the current by ${\kappa }_0^2$; an action prefactor is not fixed here.

This is a mathematical fixed-metric Maxwell-block definition. The equations are imposed on the normalized connection variations. The item does not identify ${\kappa }_{0}d\alpha$ with the geometric-normalized electrovac field of the coupled Einstein–Maxwell equations, prescribe a conversion between those normalizations, include metric variations, or claim to define the full coupled covariant phase-space current.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Fixed-topology affine sector of normalized circle connections

Local potentials and smooth circle-bundle connection data

Variation of the Lorentzian Hodge map and the linearized Maxwell equation

Lorentzian exterior calculus and Hodge star

Oriented Lorentzian fields, observers and coordinate conventions

Measurable integration extends smooth density integration

The fixed-metric restriction is essential: the Hodge-variation result gives $\delta ({\star }_g\mathcal{F})={\star }_{g}f$ when the metric perturbation is zero. For nonzero metric perturbations the terms involving $h$ in the Hodge variation remain, and the full Einstein–Maxwell current needs additional gravitational and mixed contributions. The current formula is bilinear and antisymmetric by the symmetry of the Lorentzian two-form pairing.